Let us consider a simple model for an atom. Suppose an atom is a big
ball of radius of uniform negative charge with total charge ,
with a small, heavy, point like positive charge inside. Using
Gauss's Law to find the electric field on all space due to the negative
charge, we find:

(23)

outside the ball, and

(24)

inside the ball. The force on the ``nucleus'' is thus attractive but
getting weaker outside the ball and increases linearly with its
displacement from the nucleus on the inside.

Hmmm, given a linear attractive restoring force, we can guess that the
nucleus will oscillate harmonically in the negative ball if released
from rest on the inside. Cool. Later we'll think about what happens if
it is driven harmonically at its resonant frequency. We can also
see that if we put the whole ``atom'' in an external electric field, it
will polarize, that is, become a dipole. The nucleus will be
displaced in the direction of the external field until the net force
pulling it back to the origin vanishes.

In a field stronger than , though, the nucleus is pushed
outside of the ball where the force suddenly drops off. It is ripped
away from its negative ball (or vice versa, actually, since the negative
electrons are really the lighter things). The point is, that any
material, be it insulator or conductor, can be put in a field strong
enough to rip its constituent atoms apart. They ionize, and generally
when that happens an insulator becomes a very hot conductor. This is
the phenomenon know as ``dielectric breakdown''.

Even the best insulators - glass, quartz, mica - will arc across if
placed in a big enough field. Insulators, in fact, aren't. They are
only insulating in the weak field regime, and tend to cross over from
that weak field regime to a conducting state catastrophically. A bolt
of lightning, a static spark, St. Elmo's fire, all are the result of an
insulating material being torn apart by a strong field so that it
conducts. This is one of many reasons high voltage power supplies are
so dangerous - 16,000 volts of potential difference will arc right
across a gap of a centimeter or so of air. You don't actually have to
touch the conductor to die, especially if there are any sharp points
nearby.

This is also how lightning rods work. Lightning rods don't attract
lightning (which is catastrophically destructive - you do not
want to be hit by lightning or to even have lightning hit a lightning
rod connected to your house, unless you aren't fond of your house and
want to collect the insurance). Lightning rods prevent lightning.
Let's see how.

Suppose a big old positively charged up cloud passes by overhead with no
lighting rods present. It attracts electrons from the nearby conducting
ground, and they pile up directly beneath it. They strain to reach the
cloud, running up things like our salty bodies or the trunks of nearby
tries to get a bit closer to the positive charge there. However, they
cannot easily jump over to the cloud with all that insulating air in the
way and the field grows faster than the cloud is neutralized.
Eventually the field strength gets to be great enough that the air right
above a high point starts to ionize, and BAM, current flows which
heats the air which knocks off electrons which conduct current wich
flows, further heating the air, reaching ever higher to the cloud. In
an instant, a lot of charge flows from the ground to the cloud,
which becomes neutralized. The tree, human, air, and anything else
granting passage to the huge blast of hot current sizzles gently in its
passing, dead, exploded, burnt. Air rushes back into the resulting
vacuum, creating the roll of thunder.

NOW suppose there is a lightning rod handy. When the cloud rolls by,
electrons are pulled up to the sharp tip of the lightning rod. There
even a small surplus charge is enough to cause air around the tip
to ionize, and the air around the tip begins to conduct (and will often
glow as electrons are knocked about and recombine with atoms). This
happens slowly though, slowly enough that the charged air can
spray upwards and create a conducting path to the cloud without breaking
down all at once or getting so hot. On a good day, the cloud is
neutralized without ever discharging a bolt of lightning. On a bad day,
lightning hits your lightning rod and it explodes in a shower of molten
metal and burns down your house.

Clearly lots of lightning rods are better than just one. So are
lots of trees.

Dielectric breakdown and the phenomenon of polarization will be very
important to us in the next chapter.

To conclude, may I suggest that you review the methods covered for
evaluating the potential from the field (using Gauss's Law, to find the
field) as this will be of the greatest use to us in our discussion of
capacitance. However, you should all be able to find the potential
inside a uniform ball of sphere, for example, as well.